Vanishing of the James brace product before rationalization for two classes of fibrations

Determine whether the James brace product { , }_s vanishes identically in the original (non-rationalized) fibration in each of the following settings: (i) an S^n-bundle S^n ↪ E → X with a homotopy section s, where X is simply connected and n > 1 is odd; and (ii) an S^k-bundle S^k ↪ E → G with a homotopy section s, where G is a simply connected compact Lie group and k > 1. Concretely, ascertain whether {α, β}_s = 0 for all α ∈ π_i(X) and β ∈ π_j(S^n) in case (i), and for all α ∈ π_i(G) and β ∈ π_j(S^k) in case (ii), without passing to rationalization.

Background

In Section 5 the authors analyze rationalized fibrations and show that, in certain classes, the James brace product vanishes after rationalization. Specifically, they consider (i) the rationalized fibration of any S^n-bundle over a simply connected space X with a section, for n > 1 odd, and (ii) the rationalized fibration of any S^k-bundle over a simply connected compact Lie group G with a section, for k > 1. In both situations they demonstrate vanishing of the brace product in the rationalized setting.

However, they explicitly note that they do not know whether this vanishing extends to the original (non-rationalized) fibrations. The open problem asks for a determination of whether the James brace product vanishes identically at the fibration level itself in these two classes, i.e., without invoking rationalization.